Statement of the equation

Poisson's equation is

Δφ=f{\displaystyle \Delta \varphi =f}

where Δ{\displaystyle \Delta } is the Laplace operator, and f{\displaystyle f} and φ{\displaystyle \varphi } are real or complex-valued functions on a manifold. Usually, f{\displaystyle f} is given and φ{\displaystyle \varphi } is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as

Poisson's equation may be solved using a Green's function; a general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.

Newtonian gravity

In the case of a gravitational field g due to an attracting massive object of density ρ, Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity,

∇⋅g=−4πGρ.{\displaystyle \nabla \cdot {\mathbf {g} }=-4\pi G\rho ~.}

Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential Φ,

Electrostatics

One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potentialφ for a given charge distribution ρf{\displaystyle \rho _{f}}.

The mathematical details behind Poisson's equation in electrostatics are as follows (SI units are used rather than Gaussian units, which are also frequently used in electromagnetism).

which is Coulomb's law of electrostatics. (For historic reasons, and unlike gravity's model above, the 4π{\displaystyle 4\pi } factor appears here and not in Gauss's law.)

The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing φ is no longer sufficient to calculate E, since E also depends on the magnetic vector potentialA, which must be independently computed. See Maxwell's equation in potential formulation for more on φ and A in Maxwell's equations and how Poisson's equation is obtained in this case.

as one would expect. Furthermore, the erf function approaches 1 extremely quickly as its argument increases; in practice for r > 3σ the relative error is smaller than one part in a thousand.

Surface reconstruction

Surface reconstruction is an inverse problem. The goal is to digitally reconstruct a smooth surface based on a large number of points pi (a point cloud) where each point also carries an estimate of the local surface normalni.[2] Poisson's equation can be utilized to solve this problem with a technique called Poisson Surface Reconstruction first published in (Kazhdan et al., 2006)[3]

The goal of this technique is to reconstruct an implicit functionf whose value is zero at the points pi and whose gradient at the points pi equals the normal vectors ni. The set of (pi, ni) is thus modeled as a continuous vector field V. The implicit function f is found by integrating the vector field V. Since not every vector field is the gradient of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field V to be the gradient of a function f is that the curl of V must be identically zero. In case this condition is difficult to impose, it is still possible to perform a least-squares fit to minimize the difference between V and the gradient of f.

In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field V. The basic approach is to bound the data with a finite difference grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform [trilinear interpolation] on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ni onto the nodes of the particular staggered grid cell containing pi. In (Kazhdan et al., 2006),[4] the authors give a more accurate method of discretization using an adaptive finite difference grid, i.e. the cells of the grid are smaller (the grid is more finely divided) where there are more data points. They suggest implementing this technique with an adaptive octree.

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